#### Title of Oral/Poster Presentation

Encryption, Elliptic Curves, and the Symmetries of Differential Equations

#### Class

Article

#### Graduation Year

2018

#### College

College of Science

#### Department

Mathematics and Statistics Department

#### Faculty Mentor

Andreas Malmendier

#### Presentation Type

Oral Presentation

#### Abstract

In cryptography, encryption is the process of encoding messages in such a way that only authorized parties can access them. The intended information, referred to as plaintext, is encrypted using an encryption algorithm, generating ciphertext that can only be read if decrypted. Public key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner. In a public key encryption system, any person can encrypt a message using the public key, but such a message can be decrypted only with the private key. Elliptic curve cryptography (ECC) is a particularly powerful approach to public-key cryptography based on tori or more precisely elliptic curves.

The purpose of this talk is to discuss the mathematics employed in elliptic curve encryption which is based on the algebraic structure of elliptic curves, in particular on the ability to add points. Such group structure on a torus is evident if we represent it as a fundamental domain in the complex plane with its edges identified. Once the group structure has been defined in the complex plane, the group structure on a torus is evident. In turn, an elliptic curve is parameterized over the complex plane by the Weierstrass elliptic function. Moreover, the Weierstrass elliptic function allows to identify the defining quantities of a torus with those of an elliptic curve using modular forms.

#### Location

Room 208

#### Start Date

4-13-2017 1:30 PM

#### End Date

4-13-2017 2:45 PM

Encryption, Elliptic Curves, and the Symmetries of Differential Equations

Room 208

In cryptography, encryption is the process of encoding messages in such a way that only authorized parties can access them. The intended information, referred to as plaintext, is encrypted using an encryption algorithm, generating ciphertext that can only be read if decrypted. Public key cryptography, or asymmetric cryptography, is any cryptographic system that uses pairs of keys: public keys which may be disseminated widely, and private keys which are known only to the owner. In a public key encryption system, any person can encrypt a message using the public key, but such a message can be decrypted only with the private key. Elliptic curve cryptography (ECC) is a particularly powerful approach to public-key cryptography based on tori or more precisely elliptic curves.

The purpose of this talk is to discuss the mathematics employed in elliptic curve encryption which is based on the algebraic structure of elliptic curves, in particular on the ability to add points. Such group structure on a torus is evident if we represent it as a fundamental domain in the complex plane with its edges identified. Once the group structure has been defined in the complex plane, the group structure on a torus is evident. In turn, an elliptic curve is parameterized over the complex plane by the Weierstrass elliptic function. Moreover, the Weierstrass elliptic function allows to identify the defining quantities of a torus with those of an elliptic curve using modular forms.